A Trainable Graph Combination Scheme for Belief Propagation

1
A Trainable Graph Combination Scheme for Belief
Propagation

• Kai Ju Liu
• January 11, 2006
• Advisor Davi Geiger

2
Images
3
Pairwise Markov Random Field

• Basic structure graph of vertices connected by
  edges
• Each vertex has
• observed value y
• set of possible states X
• Goal to state something about vertex states
  given observed data

4
Pairwise MRF Compatibility Functions

• Statistical dependency between yi and xi
• fi(xi, yi)
• Relation between neighboring vertices i and j
• yij(xi, xj)

5
Pairwise MRF Joint Probability

• Consider MRF with n vertices
• Proportional to product of compatibility
  functions
• Yields most likely set of states xi

6
Pairwise MRF Marginal Probability

• Yields vertex is most likely state xi
• Allows average over unknown states
• Complexity exponential in number of vertices

7
Pairwise MRF Probability Inefficiencies

• Joint probabilities share common terms
• Probability calculations can be much more
  efficient

8
Belief Propagation

• Accumulate probability calculations at vertices
• Beliefs replace probabilities

9
BP Messages

• mji(xi), message vertex j passes to vertex i
• How likely vertex j thinks, given knowledge from
  ancestors, that vertex i is in state xi

10
BP Applications

• Medical diagnoses
• Symptoms to diseases
• Insurance policies
• Various factors to risk assessment

11
BP Concerns

• When can we calculate beliefs exactly?
• When do beliefs equal probabilities?
• When is belief propagation efficient?

12
BP Singly-Connected Graphs (SCGs)

• Graphs without loops
• Messages terminate at leaf vertices
• Beliefs equal probabilities

13
BP SCG Complexity

• S states per vertex
• Marginal probabilities 8S5 multiplications, 5S5
  additions
• Beliefs 16S2 multiplications, 8S2 additions

14
BP Loopy Graphs

• Majority of useful graphs are loopy
• Message update rules are circular

15
BP Energy

• Boltzmanns equation converts probability to
  energy
• Search for energy approximations

16
Standard Belief Propagation

• Based on Bethe approximation to free energy
• Follows belief propagation rules
• Initialize messages to 1.0
• Update self-consistently until convergence

17
Generalized Belief Propagation

• Based on Kikuchi approximation to free energy
• Passes messages between regions

18
BP-TwoGraphs Goals

• Utilize properties of SCGs
• Accurate and efficient on loopy graphs

19
BP-TwoGraphs SCGs

• Consider loopy graph with n vertices
• Select two sets of SCGs that approximate the
  graph

20
BP-TwoGraphs Combining SCGs

• Calculate beliefs on each set of SCGs
• biG(xi) and biH(xi)
• Select maximum beliefs from both sets

21
BP-TwoGraphs Image Segmentation SCGs

• Rectangular grid of pixel vertices
• Horizontal graphs
• Vertical graphs
• Belief propagation exact on these graphs

22
Test Case Image Segmentation

• Classic, difficult problem
• Test images generated from artificial image

23
Test Case BP-TwoGraphs Compatibility Functions

• f compatibility function must distinguish dark
  from light
• No grayscale level discounted

24
Test Case BP-TwoGraphs Compatibility Functions

• y function controls tendency for neighboring
  vertices to be in same state
• Distinguishes only between similar and dissimilar
  states

25
Test Case Max-Flow

• Calculate maximum flows or minimum cuts of flow
  networks
• Flow network similar to pairwise MRF of
  BP-TwoGraphs
• Additional source and sink/target vertices

26
Test Case Max-Flow Capacities

• Linear source-to-vertex and vertex-to-sink
  capacities similar to f compatibility function
• No vertex disconnected from source or sink
• Capacity between pixel vertices m

27
Test Case Setup
Parameter Min. Max. Delta
Test Images Std. dev. 1.0 170.0 1.0
BP-TwoGraphs STICKINESS 0.5 1.0 0.0001
Max-Flow m 0.0 255.0 0.5
28
Test Case Results
29
Combined Graph Analysis
30
Combined Graph Analysis
31
Additional Comparisons I
32
Additional Comparisons II
33
Boundary-Based Image Segmentation

• Scale
• Image measurements
• Wavelets
• Windows of pixels

34
Boundary-Based Image Segmentation Window Vertices

• Square 2-by-2 window of pixels
• Each pixel has two states
• foreground
• background

35
Boundary-Based Image Segmentation Overlap
36
Boundary-Based Image Segmentation Graph
37
Boundary-Based Image Segmentation f

• Square l-by-l window vertices
• Each pixel has two states
• foreground
• background

38
Boundary-Based Image Segmentation y

• Scan rows/columns for directed pairs of
  neighboring states
• Rotate through 90, 180, 270 degrees

39
Boundary-Based Image Segmentation Training Images
40
Boundary-Based Image Segmentation Eagle Results
41
Boundary-Based Image Segmentation Gorilla Results
42
Conclusion

• BP-TwoGraphs
• Accurate and efficient
• Trainable parameters
• Extensive use of beliefs
• Future work
• Learning
• 3D data
• General graphs